Methods and systems using norm approximation for maximum likelihood MIMO decoding

ABSTRACT

Certain embodiments of the present disclosure provide techniques for approximate computation of l 2  norms as a part of the maximum likelihood (ML) detection: tri-maxmin, maxsum and sortsum algorithms. The proposed approximation schemes show better accuracy than conventional approximation schemes—the abssum and maxmin algorithms, while the computational complexity is very similar. The error rate performance of the ML detection that utilizes proposed norm-approximation techniques are very close to the reference ML detection with exact calculation of l 2  norms, while the computational complexity is significantly smaller.

CLAIM OF PRIORITY

This application claims the benefit of priority from U.S. ProvisionalPatent Application Ser. No. 61/105,008, filed Oct. 13, 2008 and entitled“Methods and systems for efficient norm approximation for maximumlikelihood MIMO decoder,” which is fully incorporated herein byreference for all purposes.

TECHNICAL FIELD

The present disclosure generally relates to communication, and morespecifically to methods for approximate computation of metrics as a partof the maximum likelihood decoder.

SUMMARY

Certain embodiments provide a method for decoding data transmitted on aplurality of spatial channels in a multiple-input multiple-outputwireless communications system. The method generally includescalculating non-squared norm metrics for a set of hypothesized symbolsthat correspond to coded bits transmitted over the spatial channels,searching the non-squared norm metrics for optimum metrics, squaringoptimum metrics found during the searching to obtain post-squaredmetrics, and calculating log-likelihood ratio values for coded bitstransmitted over the spatial channels using post-squared metrics.

Certain embodiments provide an apparatus for decoding data transmittedon a plurality of spatial channels in a multiple-input multiple-outputwireless communications system. The apparatus generally includes logicfor calculating non-squared norm metrics for a set of hypothesizedsymbols that correspond to coded bits transmitted over the spatialchannels, logic for searching the non-squared norm metrics for optimummetrics, logic for squaring optimum metrics found during the searchingto obtain post-squared metrics, and logic for calculating log-likelihoodratio values for coded bits transmitted over the spatial channels usingpost-squared metrics.

Certain embodiments provide an apparatus for decoding data transmittedon a plurality of spatial channels in a multiple-input multiple-outputwireless communications system. The apparatus generally includes meansfor calculating non-squared norm metrics for a set of hypothesizedsymbols that correspond to coded bits transmitted over the spatialchannels, means for searching the non-squared norm metrics for optimummetrics, means for squaring optimum metrics found during the searchingto obtain post-squared metrics, and means for calculating log-likelihoodratio values for coded bits transmitted over the spatial channels usingpost-squared metrics.

Certain embodiments provide a computer-program product for decoding datatransmitted on a plurality of spatial channels in a multiple-inputmultiple-output wireless communications system, comprising a computerreadable medium having instructions stored thereon, the instructionsbeing executable by one or more processors. The instructions generallyinclude instructions for calculating non-squared norm metrics for a setof hypothesized symbols that correspond to coded bits transmitted overthe spatial channels, instructions for searching the non-squared normmetrics for optimum metrics, instructions for squaring optimum metricsfound during the searching to obtain post-squared metrics, andinstructions for calculating log-likelihood ratio values for coded bitstransmitted over the spatial channels using post-squared metrics.

BRIEF DESCRIPTION OF THE DRAWINGS

So that the manner in which the above-recited features of the presentdisclosure can be understood in detail, a more particular description,briefly summarized above, may be had by reference to embodiments, someof which are illustrated in the appended drawings. It is to be noted,however, that the appended drawings illustrate only certain typicalembodiments of this disclosure and are therefore not to be consideredlimiting of its scope, for the description may admit to other equallyeffective embodiments.

FIG. 1 illustrates an example wireless communication system, inaccordance with certain embodiments of the present disclosure.

FIG. 2 illustrates various components that may be utilized in a wirelessdevice in accordance with certain embodiments of the present disclosure.

FIG. 3 illustrates an example transmitter and an example receiver thatmay be used within a wireless communication system in accordance withcertain embodiments of the present disclosure.

FIG. 4 illustrates a block diagram of a generic multiple-inputmultiple-output Orthogonal Frequency Division Multiplexing (MIMO-OFDM)wireless system in accordance with certain embodiments of the presentdisclosure.

FIG. 5 shows a block diagram of a maximum likelihood detector withpost-squaring compensation (PC-MLD) in accordance with certainembodiments of the present disclosure.

FIG. 6 shows a process of performing the maximum likelihood detectionwith approximate computation of l₂ norms and post-squaring compensationin accordance with certain embodiments of the present disclosure.

FIG. 6A illustrates example components capable of performing theoperations illustrated in FIG. 6.

FIG. 7 shows a complexity comparison between several approximatealgorithms for computation of norms in accordance with certainembodiments of the present disclosure.

FIG. 8 shows recommended values of tuning factors and corresponding meansquare errors at signal-to-noise ratio (SNR) value of 0 dB for differentapproximate algorithms for computation of norms in accordance withcertain embodiments of the present disclosure.

FIG. 9 shows an error rate performance loss in dB units of the PC-MLDwith different approximate algorithms for computation of norms forPedestrian A channel, 64-QAM modulation and at a packet error rate of10-2 in accordance with certain embodiments of the present disclosure.

FIG. 10 shows an error rate performance loss in dB units of the PC-MLDwith different approximate algorithms for computation of norms forPedestrian A channel, 16-QAM modulation and at a packet error rate of10-2 in accordance with certain embodiments of the present disclosure.

FIG. 11 shows an error rate performance loss in dB units of the PC-MLDwith different approximate algorithms for computation of norms forPedestrian A channel, QPSK modulation and at a packet error rate of 10-2in accordance with certain embodiments of the present disclosure.

DETAILED DESCRIPTION

The word “exemplary” is used herein to mean “serving as an example,instance, or illustration.” Any embodiment described herein as“exemplary” is not necessarily to be construed as preferred oradvantageous over other embodiments.

A maximum likelihood (ML) multiple-input multiple output (MIMO) decodingscheme requires “squared l₂ norm” as a metric for calculation oflog-likelihood ratios (LLRs) of transmitted coded bits. A computationalcomplexity of direct calculation of squared l₂ norms may be enormous,especially for high order modulation schemes and/or high dimensionalMIMO systems (i.e. high number of transmit antennas). In order to reducethe computational complexity of LLR calculation, a “non-squared l₂norms” can be utilized as the metric for obtaining LLRs of coded bits.The non-squared l₂ norms can be converted to the squared l₂ norms byapplying a post squaring operations, which results into the ML detectionwith post-squaring compensation (the PC-MLD technique).

However, the computational complexity of obtaining exact non-squared l₂norms may be higher than that of obtaining squared l₂ norms due toutilized square-root operations. On the other hand, computation of l₂norms may be less complex than that of squared l₂ norms if someapproximate algorithm is applied that does not utilize square-rootoperations.

Exemplary Wireless Communication System

The techniques described herein may be used for various broadbandwireless communication systems, including communication systems that arebased on an orthogonal multiplexing scheme. Examples of suchcommunication systems include Orthogonal Frequency Division MultipleAccess (OFDMA) systems, Single-Carrier Frequency Division MultipleAccess (SC-FDMA) systems, and so forth. An OFDMA system utilizesorthogonal frequency division multiplexing (OFDM), which is a modulationtechnique that partitions the overall system bandwidth into multipleorthogonal sub-carriers. These sub-carriers may also be called tones,bins, etc. With OFDM, each sub-carrier may be independently modulatedwith data. An SC-FDMA system may utilize interleaved FDMA (IFDMA) totransmit on sub-carriers that are distributed across the systembandwidth, localized FDMA (LFDMA) to transmit on a block of adjacentsub-carriers, or enhanced FDMA (EFDMA) to transmit on multiple blocks ofadjacent sub-carriers. In general, modulation symbols are sent in thefrequency domain with OFDM and in the time domain with SC-FDMA.

One specific example of a communication system based on an orthogonalmultiplexing scheme is a WiMAX system. WiMAX, which stands for theWorldwide Interoperability for Microwave Access, is a standards-basedbroadband wireless technology that provides high-throughput broadbandconnections over long distances. There are two main applications ofWiMAX today: fixed WiMAX and mobile WiMAX. Fixed WiMAX applications arepoint-to-multipoint, enabling broadband access to homes and businesses,for example. Mobile WiMAX offers the full mobility of cellular networksat broadband speeds.

IEEE 802.16x is an emerging standard organization to define an airinterface for fixed and mobile broadband wireless access (BWA) systems.These standards define at least four different physical layers (PHYs)and one media access control (MAC) layer. The OFDM and OFDMA physicallayer of the four physical layers are the most popular in the fixed andmobile BWA areas respectively.

FIG. 1 illustrates an example of a wireless communication system 100 inwhich embodiments of the present disclosure may be employed. Thewireless communication system 100 may be a broadband wirelesscommunication system. The wireless communication system 100 may providecommunication for a number of cells 102, each of which is serviced by abase station 104. A base station 104 may be a fixed station thatcommunicates with user terminals 106. The base station 104 mayalternatively be referred to as an access point, a Node B or some otherterminology.

FIG. 1 depicts various user terminals 106 dispersed throughout thesystem 100. The user terminals 106 may be fixed (i.e., stationary) ormobile. The user terminals 106 may alternatively be referred to asremote stations, access terminals, terminals, subscriber units, mobilestations, stations, user equipment, etc. The user terminals 106 may bewireless devices, such as cellular phones, personal digital assistants(PDAs), handheld devices, wireless modems, laptop computers, personalcomputers, etc.

A variety of algorithms and methods may be used for transmissions in thewireless communication system 100 between the base stations 104 and theuser terminals 106. For example, signals may be sent and receivedbetween the base stations 104 and the user terminals 106 in accordancewith OFDM/OFDMA techniques. If this is the case, the wirelesscommunication system 100 may be referred to as an OFDM/OFDMA system.

A communication link that facilitates transmission from a base station104 to a user terminal 106 may be referred to as a downlink (DL) 108,and a communication link that facilitates transmission from a userterminal 106 to a base station 104 may be referred to as an uplink (UL)110. Alternatively, a downlink 108 may be referred to as a forward linkor a forward channel, and an uplink 110 may be referred to as a reverselink or a reverse channel.

A cell 102 may be divided into multiple sectors 112. A sector 112 is aphysical coverage area within a cell 102. Base stations 104 within awireless communication system 100 may utilize antennas that concentratethe flow of power within a particular sector 112 of the cell 102. Suchantennas may be referred to as directional antennas.

FIG. 2 illustrates various components that may be utilized in a wirelessdevice 202 that may be employed within the wireless communication system100. The wireless device 202 is an example of a device that may beconfigured to implement the various methods described herein. Thewireless device 202 may be a base station 104 or a user terminal 106.

The wireless device 202 may include a processor 204 which controlsoperation of the wireless device 202. The processor 204 may also bereferred to as a central processing unit (CPU). Memory 206, which mayinclude both read-only memory (ROM) and random access memory (RAM),provides instructions and data to the processor 204. A portion of thememory 206 may also include non-volatile random access memory (NVRAM).The processor 204 typically performs logical and arithmetic operationsbased on program instructions stored within the memory 206. Theinstructions in the memory 206 may be executable to implement themethods described herein.

The wireless device 202 may also include a housing 208 that may includea transmitter 210 and a receiver 212 to allow transmission and receptionof data between the wireless device 202 and a remote location. Thetransmitter 210 and receiver 212 may be combined into a transceiver 214.A plurality of transmit antennas 216 may be attached to the housing 208and electrically coupled to the transceiver 214. The wireless device 202may also include (not shown) multiple transmitters, multiple receivers,and multiple transceivers.

The wireless device 202 may also include a signal detector 218 that maybe used in an effort to detect and quantify the level of signalsreceived by the transceiver 214. The signal detector 218 may detect suchsignals as total energy, energy per subcarrier per symbol, powerspectral density and other signals. The wireless device 202 may alsoinclude a digital signal processor (DSP) 220 for use in processingsignals.

The various components of the wireless device 202 may be coupledtogether by a bus system 222, which may include a power bus, a controlsignal bus, and a status signal bus in addition to a data bus.

FIG. 3 illustrates an example of a transmitter 302 that may be usedwithin a wireless communication system 100 that utilizes OFDM/OFDMA.Portions of the transmitter 302 may be implemented in the transmitter210 of a wireless device 202. The transmitter 302 may be implemented ina base station 104 for transmitting data 306 to a user terminal 106 on adownlink 108. The transmitter 302 may also be implemented in a userterminal 106 for transmitting data 306 to a base station 104 on anuplink 110.

Data 306 to be transmitted is shown being provided as input to MIMOencoder 312. The MIMO encoder may encode the data 306 and map them ontoM constellation points. The mapping may be done using some modulationconstellation, such as binary phase-shift keying (BPSK), quadraturephase-shift keying (QPSK), 8 phase-shift keying (8PSK), quadratureamplitude modulation (QAM), etc, or null (zero valued) modulation if thesub-carrier is not assigned. Thus, the MIMO encoder 312 may outputN_(FFT) parallel symbol streams 316 per antenna path, each symbol stream316 corresponding to N_(FFT) orthogonal subcarriers of the inverse fastFourier transforms (IFFTs) 320. These N_(FFT) parallel symbol streams316 are represented in the frequency domain and may be converted intoN_(FFT) parallel time domain sample streams 318 by IFFT components 320.

A brief note about terminology will now be provided. N_(T)×N_(FFT)parallel modulations in the frequency domain are equal to N_(T)×N_(FFT)modulation symbols in the frequency domain, which are equal to N_(T)parallel N_(FFT) mapping and N_(T) parallel N_(FFT)-point IFFTs in thefrequency domain, which is equal to N_(T) (useful) OFDM symbols in thetime domain, each of which is equal to N_(FFT) samples in the timedomain, where N_(T) is the number of transmit antennas 330. Afterinserting Guard samples, one OFDM symbol in the time domain, NS, isequal to NCP (the number of guard samples per OFDM symbol)+N_(FFT) (thenumber of useful samples per OFDM symbol) samples.

For one of N_(T) antenna path, the N_(FFT) parallel time domain samplestreams 318 may be converted into an OFDM/OFDMA symbol stream 322 byparallel-to-serial (P/S) converter 324. Guard insertion component 326may insert guard interval between successive OFDM/OFDMA symbols in theOFDM/OFDMA symbol stream 322 generating N_(S)=N_(CP)+N_(FFT) samples.The signal from the guard insertion component 326 may then beup-converted to a desired transmit frequency band by a radio frequency(RF) front end component 328, and the antenna array 330 may thentransmit the resulting signal 332 across multiple spatial subchannels334.

FIG. 3 also illustrates an example of a receiver 304 that may be usedwithin a wireless device 202 that utilizes OFDM/OFDMA. Portions of thereceiver 304 may be implemented in the receiver 212 of a wireless device202. The receiver 304 may be implemented in a user terminal 106 forreceiving data 306 from a base station 104 on a downlink 108. Thereceiver 304 may also be implemented in a base station 104 for receivingdata 306 from a user terminal 106 on an uplink 110.

The transmitted signal 332 is shown traveling over a plurality ofspatial subchannels 334. When a signal 332′ is received by the antennaarray 330′, the received signal 332′ may be downconverted to a basebandsignal by RF front end components 328′. Guard removal components 326′may then remove the guard intervals that were inserted betweenOFDM/OFDMA symbols by the guard insertion components 326.

For one of N_(T) antenna path, the output of the guard removal component326′ may be provided to S/P converter 324′. The S/P converter 324′ maydivide the OFDM/OFDMA symbol stream 322′ into the N_(FFT) paralleltime-domain symbol streams 318′, each of which corresponds to one of theN_(FFT) orthogonal subcarriers. Fast Fourier transform (FFT) component320′ may convert the N_(FFT) parallel time-domain symbol streams 318′into the frequency domain and output N_(FFT) parallel frequency-domainsymbol streams 316′.

A MIMO decoder 312′ may perform the inverse of the encoding and symbolmapping operations that were performed by the MIMO encoder 312 therebyoutputting the same number of data stream 306′ as the data 306. Ideally,this data stream 306′ corresponds to the data 306 that was provided asinput to the transmitter 302. Note that elements 312′, 316′, 320′, 318′and 324′ may all be found in a baseband processor 340′.

Exemplary MIMO-OFDM System Model

FIG. 4 shows a block diagram of a generic multiple-input multiple-output(MIMO) OFDM wireless communication system with N_(t) transmit and N_(r)receive antennas. The system model for the kth sub-carrier (frequencysubchannel) may be represented with linear equation:y _(k) =H _(k) x _(k) +n _(k), k=1, 2, . . . ,N_(FFT),  (1)where N_(FFT) is the number of orthogonal sub-carriers (frequency bins)in MIMO-OFDM system.

In equations and accompanying disclosure below, the sub-carrier index kis omitted for simplicity. Therefore, the system model can be re-writtenusing simpler notation as:

$\begin{matrix}{{y = {{Hx} + n}},} & (2) \\{{y = \lbrack {y_{1}\mspace{14mu} y_{2}\mspace{14mu}\ldots\mspace{14mu} y_{N_{r}}} \rbrack^{T}},} & (3) \\{{H = {\lbrack {h_{1}\mspace{14mu} h_{2}\mspace{14mu}\ldots\mspace{14mu} h_{N_{t}}} \rbrack = \begin{bmatrix}{h_{11}\mspace{14mu} h_{12}\mspace{14mu}\ldots\mspace{14mu} h_{1N_{t}}} \\\ldots \\{h_{N_{r}1}\mspace{14mu} h_{N_{r}2}\mspace{14mu}\ldots\mspace{14mu} h_{N_{r}N_{t}}}\end{bmatrix}}},} & (4)\end{matrix}$where y is [N_(r)×1] vector of received symbols, H is [N_(r)×N_(t)]matrix of channel gains and h_(j) is its jth column vector that containschannel gains between the transmit antenna j and Nr receive antennas, xis [N_(t)×1] transmitted symbol vector, n is [N_(r)×1] complex noisevector with covariance matrix E(nn^(H)).

As illustrated in FIG. 4, the transmission signal may be first encodedby the MIMO encoder 410. A redundancy may be included to protect theinformation data during the transmission over noisy wireless channels.The encoded signal may then be split into N_(t) spatial data streams x₁,x₂, . . . , x_(N) _(t) , as shown in FIG. 4. A plurality of spatial datastreams can be converted into a time domain by utilizing the InverseFast Fourier Transform (IFFT) units 412 ₁, . . . , 412 _(N) _(t) . Thesignal may then be up-converted to a desired transmission frequency bandand transmitted from N_(t) transmit antennas 414 ₁, . . . , 414 _(N)_(t) over N_(r)·N_(t) single-input single-output (SISO) channels.

Nr receive antennas 416 ₁, . . . , 416 _(N) _(r) are employed at thereceiver. Received data streams can be converted back into a frequencydomain by using the Fast Fourier Transform (FFT) units 418 ₁, . . . ,418 _(N) _(r) . A frequency domain signal may be input into a MIMOdetector 420 that generates reliability messages for coded bitstransmitted over a plurality of spatial subchannels. A reliabilitymessage represents a probability that the particular transmitted codedbit is either bit “0” or bit “1”. This information can be utilized bythe outer MIMO channel decoder 422, and the estimated information data{circumflex over (x)} for a plurality of spatial subchannels (transmitantennas) are available after removing the redundancy included at thetransmitter.

Exemplary Vector NORM

For a given vector v, l_(p) norm can be defined as:

$\begin{matrix}{v = \lbrack {v_{1}\mspace{14mu} v_{2}\mspace{14mu}\ldots\mspace{14mu} v_{j}\mspace{14mu}\ldots\mspace{14mu} v_{N}} \rbrack^{T}} & (7) \\\begin{matrix}{l_{p} = {v}_{p}} \\{= ( {{v_{1}}^{p} + {{v_{2}}^{p}\mspace{14mu}\ldots} + {v_{N}}^{p}} )^{1/p}} \\{{= ( {\sum\limits_{j = 1}^{N}{v_{j}}^{p}} )^{1/p}},}\end{matrix} & (8)\end{matrix}$where p is a real number with p≧1, and the scalar element v_(j) can beeither real or complex number.

The squared l_(p) norm may be defined as follows:Squared l _(p) =sl _(p) =l _(p) ² =∥v∥ _(p) ².  (9)The squared l₂ norms (parameter p=2) can be utilized for a metriccomputation as a part of the maximum likelihood detection.Exemplary Maximum Likelihood Detection

The maximum likelihood (ML) detection is well-known technique in theart, which utilizes maximum a posteriori (MAP) or equivalent Log-MAPalgorithm to determine the most likely transmitted modulation symbols.The ML detection can achieve optimal accuracy because it evaluates allmodulation symbols that can be potentially transmitted. The Log-MAPdetector uses log likelihood ratios (LLRs) of coded bits as reliabilityinformation to decide whether a bit “0” or a bit “1” is communicatedbetween a transmitter and a receiver over a wireless channel.

If b_(k) is the kth bit of the transmitted symbol vector x, thecorresponding LLR L(b_(k)) can be represented as follows:

$\begin{matrix}\begin{matrix}{{L( b_{k} )} = {L\; L\;{R( {b_{k}❘y} )}}} \\{= {\log\lbrack \frac{P( {b_{k} = {0❘y}} )}{P( {b_{k} = {1❘y}} )} \rbrack}} \\{= {\log\lbrack \frac{\sum\limits_{{x\text{:}b_{k}} = 0}{P( {x❘y} )}}{\sum\limits_{{x\text{:}b_{k}} = 1}{P( {x❘y} )}} \rbrack}} \\{= {\log\lbrack \frac{\sum\limits_{{x\text{:}b_{k}} = 0}{{p( {y❘x} )}{P(x)}}}{\sum\limits_{{x\text{:}b_{k}} = 1}{{p( {y❘x} )}{P(x)}}} \rbrack}} \\{= {\log\lbrack \frac{\sum\limits_{{x\text{:}b_{k}} = 0}{p( {y❘x} )}}{\sum\limits_{{x\text{:}b_{k}} = 1}{p( {y❘x} )}} \rbrack}}\end{matrix} & (10)\end{matrix}$where expression “x: b_(k)=0” denotes a set of candidate transmissionbits x with the kth information bit equal to “0”, expression “x:b_(k)=1” denotes a set of candidate transmission bits x with the kthinformation bit equal to “1”, p(x) is a probability density function ofcandidate vector x, P(x) is a probability of x, and it is assumed that xis equally distributed.

The Gaussian probability density function may be associated with thetransmission symbol vector x. In this case, expression (10) may besimplified as:

$\begin{matrix}\begin{matrix}{{L( b_{k} )} = {{LLR}( b_{k} \middle| y )}} \\{= {\log\lbrack \frac{\sum\limits_{{x\text{:}b_{k}} = 0}^{\;}{p( y \middle| x )}}{\sum\limits_{{x\text{:}b_{k}} = 1}^{\;}{p( y \middle| x )}} \rbrack}} \\{{= {\log\lbrack \frac{\sum\limits_{{x\text{:}b_{k}} = 0}^{\;}{\exp( {- {d(x)}} )}}{\sum\limits_{{x\text{:}b_{k}} = 1}^{\;}{\exp( {- {d(x)}} )}} \rbrack}},}\end{matrix} & (11)\end{matrix}$where metric d(x) may be defined as:

$\begin{matrix}\begin{matrix}{{d(x)} = {d( {x_{1},{\ldots\mspace{14mu} x_{j}\mspace{14mu}\ldots}\mspace{14mu},x_{N_{t}}} )}} \\{{= \frac{{{y - {Hx}}}^{2}}{\sigma_{n}^{2}}},}\end{matrix} & (12)\end{matrix}$where σ_(n) ² is a noise variance at the receiver. For the jth spatialdata stream, x_(j)εC^(M=2) ^(B) where M is the number of constellationpoints and B is a modulation order (number of bits per modulationsymbol). The operational complexity of the detection algorithm given byequation (11) is proportional to O(M^(N) ^(t) ) and corresponds to thenumber of hypothesized transmission symbol vectors x in equation (11).

In order to reduce computational complexity of the Log-MAP approach, theMax-Log-MAP algorithm may be applied. If the Gaussian probabilitydensity function for the transmission symbol vector x is again assumed,the LLR for the kth bit of the transmission signal vector x L(b_(k)) maybe computed as:

$\begin{matrix}\begin{matrix}{{L( b_{k} )} = {{LLR}( b_{k} \middle| y )}} \\{= {\log\lbrack \frac{\sum\limits_{{x\text{:}b_{k}} = 0}^{\;}{p( y \middle| x )}}{\sum\limits_{{x\text{:}b_{k}} = 1}^{\;}{p( y \middle| x )}} \rbrack}} \\{\approx {\log\lbrack \frac{\max\limits_{{x\text{:}b_{k}} = 0}^{\;}{p( y \middle| x )}}{\max\limits_{{x\text{:}b_{k}} = 1}^{\;}{p( y \middle| x )}} \rbrack}} \\{= {\log\lbrack \frac{\max\limits_{{x\text{:}b_{k}} = 0}^{\;}{\exp( {- {d(x)}} )}}{\max\limits_{{x\text{:}b_{k}} = 1}^{\;}{\exp( {- {d(x)}} )}} \rbrack}} \\{= {{\min\limits_{{x\text{:}b_{k}} = 1}{d(x)}} - {\min\limits_{{x\text{:}b_{k}} = 0}{{d(x)}.}}}}\end{matrix} & (13)\end{matrix}$The operational complexity of this approach is still proportional toO(M^(N) ^(T) ), but, as shown in expression (13), search operations maynow be used instead of summation operations of the Log-MAP approachgiven by equation (11).

As shown by equation (12), calculation of LLRs in both Log-MAP and theMax-Log-MAP techniques may be based on l₂ ² norms. Assuming unitarynoise variance at the receiver (after pre-whitening, for example), thecth metric d_(c) from equations (11)-(13) may be represented as:

d _(c) =l ₂ ² =∥v∥ ₂ ²where, v=y−Hx, c=1, 2, . . . ,M ^(N) ^(t) .  (14)Exemplary Maximum Likelihood Dection with Non-Squared NORMS

The Max-Log-MAP ML detection with post-squaring compensation of norms(hereinafter abbreviated as the PC-MLD) may be applied to reduce acomputational complexity of the conventional Max-Log-MAP MLD algorithmwhile preserving the error-rate performance of the conventional MLdetection. In the PC-MLD approach, the transmission signal may behypothesized based on non-squared l₂ norms instead of squared l₂ norms,and then the searching for minima non-squared distances may beperformed. A squaring of metrics may be postponed until minimanon-squared norms are determined. By computing non-squared norms and bypost-squaring only a limited number of calculated metrics, computationalcomplexity may be reduced, while identical detection accuracy can beachieved as for the conventional Max-Log-MAP ML solution.

Once the minima metrics for all coded bits are determined, thepost-squaring compensation of minima metrics may be performed. Thepost-squaring of minima metrics produces identical results as minimametrics of the original ML detection, i.e. l₂ ² norms are equivalent tothe post-squared l₂ norms.

From equation (14), the relationship between l₂ ² norm and l₂ norm maybe obtained as follows:

$\begin{matrix}{\begin{matrix}{d_{c} = {l_{2}^{2} = {v}_{2}^{2}}} \\{= ( \sqrt{\sum\limits_{j = 1}^{N_{t}}{v_{j}}^{2}} )^{2}} \\{= {( d_{l_{2},c} )^{2} = ( l_{2} )^{2}}}\end{matrix}{where},{v = {y - {Hx}}},{c = 1},2,\ldots\mspace{14mu},{M^{N_{t}}.}} & (15)\end{matrix}$

The post squaring of minima metric may be applied to recover squared l₂norms required for calculation of LLRs of coded bits. Let d_(l) _(d)_(,min) be the arbitrary minimum l₂ norm as a result of minima search:d _(l) ₂ _(,min)=min(d _(l) ₂ _(,c)).  (16)The post-squaring of the smallest l₂ norm results into equivalentminimum l₂ ² norm:

$\begin{matrix}\begin{matrix}{d_{\min} = ( d_{l_{2},\min} )^{2}} \\{= ( {\min\limits_{x}( {v}_{2} )} )^{2}} \\{= ( {\min\limits_{x}( \sqrt{\sum\limits_{j = 1}^{N_{t}}{v_{j}}^{2}} )} )^{2}} \\{= {\min\limits_{x}( ( \sqrt{\sum\limits_{j = 1}^{N_{t}}{v_{j}}^{2}} )^{2} )}} \\{= {\min\limits_{x}{( {v}_{2}^{2} ).}}}\end{matrix} & (17)\end{matrix}$Therefore, if no approximation for calculation of l₂ norms is utilized,then the result of minima search algorithm using l₂ norms is equivalentto the result of minima search using directly computed l₂ ² norms.

The post-squaring compensation may be applied as a part of the MLdetection with a QR preprocessing. FIG. 5 shows a block diagram of theMax-Log-MAP ML detector with post-squaring compensation and the QRpreprocessing. For every spatial data stream j=1, 2, . . . , N_(t), thechannel matrix may be permuted by unit 510 such that the rightmostcolumn of a permuted matrix corresponds to the jth decoded spatialstream. The QR decomposition of the permuted channel matrix can be thenperformed, and unitary matrix Q and upper triangular matrix R may beobtained. The vector of received symbol may be rotated by applying theunitary matrix Q, which is equivalent to the zero-forcing (ZF)filtering.

For every spatial data stream I=1, 2, . . . , N_(t), one rotated spatialdata stream may be sliced by unit 520 to obtain estimated modulationsymbols for that particular spatial data stream j. All other (N_(t)−1)spatial data streams may be hypothesized with all possible transmissionsymbols. Therefore, unit 520 calculates N_(t)·M^(N) ^(t) ⁻¹ metricsbased on l₂ norms. One approximate algorithm among a plurality ofproposed algorithms from the present disclosure may be applied tocalculate non-squared l₂ norms. Subsequently, for every transmission bitk=1, 2, . . . , N_(t)·B, unit 530 performs search for minima metrics forall signal-vector hypotheses x for which bit k is equal to bit “0”, andfor all hypotheses x for which bit k is equal to bit “1”. Computationalcomplexity of this candidate search process is therefore proportional toO(N_(t)·B·N_(t)·M^(N) ^(t) ⁻¹).

Once the minima metrics for all coded bits are determined, unit 540 mayperform the post-squaring compensation of minima metrics. As shownbefore, the post-squaring of minima metrics produces identical resultsas minima metrics of the original ML detection, i.e. directly computedl₂ ² norms are equivalent to post-squared l₂ norms if no approximationfor calculating l₂ norms is utilized. For every coded transmission bitk=1, 2, . . . , N_(t)·B, the LLR L(b_(k)) may be calculated by unit 550based on squared minima metrics as follows:

$\begin{matrix}\begin{matrix}{{L( b_{k} )} = {\log\lbrack \frac{\max\limits_{{x\text{:}b_{k}} = 0}^{\;}{\exp( {- {d(x)}} )}}{\max\limits_{{x\text{:}b_{k}} = 1}^{\;}{\exp( {- {d(x)}} )}} \rbrack}} \\{= {{\min\limits_{{x\text{:}b_{k}} = 1}{d(x)}} - {\min\limits_{{x\text{:}b_{k}} = 0}{d(x)}}}} \\{= {\log\lbrack \frac{\max\limits_{{x\text{:}b_{k}} = 0}^{\;}{\exp( {- ( {d_{l_{2}}(x)} )^{2}} )}}{\max\limits_{{x\text{:}b_{k}} = 1}^{\;}{\exp( {- ( {d_{l_{2}}(x)} )^{2}} )}} \rbrack}} \\{= {( {\min\limits_{{x\text{:}b_{k}} - 1}{d_{l_{2}}(x)}} )^{2} - {( {\min\limits_{{x\text{:}b_{k}} - 0}{d_{l_{2}}(x)}} )^{2}.}}}\end{matrix} & (18)\end{matrix}$

Assuming unitary variance of the effective noise at the receiver (afterpre-whitening, for example), the cth metric d_(l) ₂ _(,c) from themetric calculation block 520 may be determined as:d _(l) ₂ _(,c) =l ₂ =∥v∥ ₂where, v=y _(r) −Rx, c=1, 2, . . . ,N _(t) M ^(N) ^(t) ⁻¹.  (19)

Calculated LLRs for all N_(t)·B coded bits transmitted over a pluralityof spatial subchannels may be utilized by the outer channel decoder 560,which generates decoded information bits at its output.

Exemplary Techniques for Approximate Computation of Non-Squared NORMS

It can be observed from equations (8) and (19) that the ML detectionwith post-squaring compensation may require a higher computationalcomplexity than the original approach based on direct computation of l₂² norms due to square-root operations. However, this holds only if theexact non-squared l₂ norms are calculated. The computational complexityof the ML detection with post-squaring compensation may be significantlyreduced if approximate algorithms for calculation of l₂ norms areapplied that do not utilize square-root operations.

In the present disclosure, several l₂ norm approximation schemes areproposed, in particular: tri-maxmin, maxsum and sortsum approximationalgorithms. It can be shown that proposed approximate techniques forcomputation of l₂ norms show better error rate performance thanconventional approximate algorithms such as abssum or maxmin techniques.The error rate performance of the ML MIMO decoder using one of theproposed approximate schemes for computation of l₂ norms are very closeto the error rate performance of the reference ML MIMO decoder thatutilizes the exact calculation of l₂ norms. In the same time, thecomputational complexity is substantially reduced.

As an illustrative example, a wireless system with two transmit and tworeceive antennas may be assumed. Then, the following norm-vector may beconsidered according to equation (7):v=[v ₁ v ₂]^(T).  (20)The complex vector from equation (20) may also be denoted by the vectorz composed of real elements:

$\begin{matrix}\begin{matrix}{z = \lbrack {{{real}( v_{1} )},{{imag}( v_{1} )},{{real}( v_{2} )},{{imag}( v_{2} )}} \rbrack^{T}} \\{= {\lbrack {z_{1},z_{2},z_{3},z_{4}} \rbrack^{T}.}}\end{matrix} & (21)\end{matrix}$

The exact l₂ norm may be then computed as:

$\begin{matrix}\begin{matrix}{l_{2,{exact}} = {v}_{2}} \\{= \sqrt{{v_{1}}^{2} + {v_{2}}^{2}}} \\{= {\sqrt{{z_{1}}^{2} + {z_{2}}^{2} + {z_{3}}^{2} + {z_{4}}^{2}}.}}\end{matrix} & (22)\end{matrix}$

For one embodiment of the present disclosure, a tri-maxmin approximatealgorithm may be utilized for computation of l₂ norms. Equation (22) maybe rewritten as:

$\begin{matrix}\begin{matrix}{l_{2,{exact}} = \sqrt{{v_{1}}^{2} + {v_{2}}^{2}}} \\{= {\sqrt{( \sqrt{{z_{1}}^{2} + {z_{2}}^{2}} )^{2} + ( \sqrt{{z_{3}}^{2} + {z_{4}}^{2}} )^{2}}.}}\end{matrix} & (23)\end{matrix}$

Based on equation (23), l₂ norm may be approximately computed using thetri-maxmin algorithm with maxmin approximation:l _(2,t max min)=maxmin(maxmin(|z ₁ |,|z ₂|),maxmin(|z ₃ |,|z₄|)),  (24)where maxmin may be given as the following expression with tuning factorF:maxmin(a,b)=max(a,b)+F·min(a,b).  (25)The tuning factor F of 0.25 may be employed considering accuracy of theapproximate algorithms and implementation complexity.

The tuning factor F may be adjusted to control the accuracy of theproposed algorithm. For example, accuracy of the proposed tri-maxminalgorithm may be reduced if the tuning factor F is not applied inequation (25). The chosen value of the tuning factor of 0.25 may providesufficient accuracy of the proposed approximate algorithm, withrelatively low computational complexity since the multiplication by thetuning factor of 0.25 from equation (25) can be efficiently implementedwith a simple right shift operation by two bit-positions.

For another embodiment of the present disclosure, a sort-sum algorithmmay be utilized for approximate computation of l₂ norms. In the sort-sumapproach, a linear combination of sorted vector elements can be utilizedto approximate computation of l₂ norm, as given by:

$\begin{matrix}{{s = {{sort}( {{z},^{\backprime}{{descending}\mspace{14mu}{order}^{\prime}}} )}},} & (26) \\{l_{2,{sortsum}} = {\sum\limits_{j}{F_{j} \cdot s_{j}}}} & (27)\end{matrix}$where s_(j) is a sorted version of |z_(j)| in descending order. Thefollowing tuning factors may be applied considering accuracy of theapproximate algorithm and implementation complexity:F _(j)=[0.875 0.5 0.25 0.125] orF _(j)=[0.875 0.4375 0.3125 0.125].  (28)

For yet another embodiment of the present disclosure, the exactcomputation of l₂ norms given by equation (22) may be approximated byapplying a maxsum algorithm. For the sortsum approximation given byequations (26)-(27), a linear combination of sorted vector elements maybe utilized. A more simplified version of the sortsum approximationcalled the maxsum algorithm may be applied, as given by:

$\begin{matrix}{\begin{matrix}{l_{2,{maxsum}} = {{\max( {z_{j}} )} + {F \cdot ( {\sum\limits_{j,{j \neq m}}{z_{j}}} )}}} \\{= {{\max( {z_{j}} )} + {F \cdot ( {( {\sum\limits_{j}{z_{j}}} ) - {\max( {z_{j}} )}} )}}}\end{matrix}{{{where}\mspace{14mu} m} = {\underset{j}{\arg\;\max}{( {z_{j}} ).}}}} & (29)\end{matrix}$The tuning factor F of 0.25 may be utilized considering accuracy of theapproximate algorithm and implementation complexity.

FIG. 6 shows a process of the Max-Log-MAP PC-MLD with approximatecomputation of l₂ norms. At 610, for every spatial data stream j=1, 2, .. . , N_(t), the channel matrix may be permuted such that the rightmostcolumn of the permuted matrix corresponds to the jth decoded stream. TheQR decomposition of permuted channel matrix for every utilized spatialdata stream is then performed, and the vector of received symbol may berotated by applying the zero-forcing (ZF) technique.

At 620, for every spatial data stream j=1, 2, . . . , N_(t), one rotatedspatial stream may be sliced in order to obtain estimated modulationsymbols for that particular spatial data stream j. All other (N_(t)−1)spatial data streams may be hypothesized with all possible transmissionsymbols. At 630, N_(t)·M^(N) ^(t) ⁻¹ metrics based on l₂ norms may becalculated by utilizing one of the proposed approximation techniques:tri-maxmin, maxsum or sortsum algorithm. At 640, for every transmissionbit k=1, 2, . . . , N_(t)·B, a search for minima metrics may beperformed for all signal-vector hypotheses x for which bit k is equal tobit “0”, and for all hypotheses x for which bit k is equal to bit “1”.

At 650, once the minima metrics for all coded bits are determined, thepost-squaring compensation of minima metrics may be performed. At 660,for every coded transmission bit k=1, 2, . . . , N_(t)·B, LLR L(b_(k))may be calculated based on squared minima metrics as given by equation(18). At 670, calculated LLRs for all N_(t)·B coded bits transmittedover a plurality of spatial subchannels for a single frequency bin maybe passed to outer channel decoding and utilized for generating decodedinformation bits.

For the comparison purpose, an approximate computation of l₁ norm isalso provided, as well as techniques for approximate computation of l₂norm from the prior art. According to equation (8) the exact l₁ norm maybe computed as:

$\begin{matrix}\begin{matrix}{l_{1,{exact}} = {v}_{1}} \\{= {{v_{1}} + {{v_{2}}.}}}\end{matrix} & (30)\end{matrix}$Equation (30) may be rewritten as:

$\begin{matrix}\begin{matrix}{l_{1,{exact}} = {{v_{1}} + {v_{2}}}} \\{= {\sqrt{{z_{1}}^{2} + {z_{2}}^{2}} + {\sqrt{{z_{3}}^{2} + {z_{4}}^{2}}.}}}\end{matrix} & (31)\end{matrix}$Then, l₁ norm can be approximated by the sum of maxmin functions usingthe following maxmin approximation:l _(1,s max min)=maxmin(|z ₁ |,|z ₂|)+maxmin(|z ₃ |,|z ₄|).  (32)

The well-known abssum approximation may be represented as:

$\begin{matrix}{{l_{1,{abssum}}( {{or}\mspace{14mu} l_{2,{abssum}}} )} = {F \cdot {\sum\limits_{j}{{z_{j}}.}}}} & (33)\end{matrix}$

The conventional maxmin approximation with four real elements may berepresented as an extension of the maxmin approximation with two realelements:l _(2,max min)=max(|z ₁ |,|z ₂ |,|z ₃ |,|z ₄|)+F·min(|z ₁ |,|z ₂ |,|z ₃|,|z ₄|).  (34)Exemplary Determination of Tuning Factors for Approximate Computation ofNORM Metrics

As previously described, the tuning factor F may be utilized to improvethe accuracy of approximate computation of l₂ norms. The optimal tuningfactor may be determined by calculating the minimum mean square error(MMSE) between the exact norm value and the approximated norm value.

The MIMO signal from FIG. 4 can be modeled as:y=Hx+n,  (35)where y is [N_(r)×1] received symbol vector, H is [N_(r)×N_(t)] channelmatrix, x is [N_(t)×1] transmit symbol vector with covariance matrixE(x^(H)x)=1 and x is randomly distributed in a given constellationspace, and n is [N_(r)×1] complex white Gaussian noise vector withcovariance matrix E(n^(H)n).

In the case of ML decoding, squared l₂ norms may be computed for allpossible values of x as:d=∥v∥ ₂ ² =∥y−Ĥx∥ ₂ ²,  (36)where Ĥ is a matrix of estimated channel gains.

On the other hand, in the case of PC-MLD, the l₂ norm for all possiblevalues of x may be calculated as:d _(l) ₂ (x)=∥v∥ ₂ =∥y−Ĥx∥ ₂.  (37)

Assuming that the channel is known, the error vector signal model may begiven as:

$\begin{matrix}{v \approx \{ \begin{matrix}n & {{{{if}\mspace{14mu} a} = t},{x_{t} \in x},{x_{a} \in x}} \\{n + {H( {x_{t} - x_{a \neq t}} )}} & {{else},}\end{matrix} } & (38)\end{matrix}$where x_(t) is a transmission symbol vector at a given time t and x_(a)is any arbitrary symbol vector in symbol-vector x, and n is a complexwhite Gaussian noise. The second term of equation (38) in the case ofa≠t may be only considered as an offset value that represents amultiplication of a random channel matrix and the difference of any twoconstellation points. The optimal tuning factor for approximatecomputation of l₂ norms may be obtained in an effort to minimize theerror vector from equation (38).Exemplary Complexity Comparison of Different NORM ApproximationTechniques

FIG. 7 shows a complexity comparison between several schemes forapproximate computation of l₂ norms. It may be assumed that the tuningfactor is equal to 2−n, where n is an integer. In this way,implementation complexity of the approximate algorithm for computationof l₂ norms may be reduced, while the accuracy is not sacrificed.

The bit width and memory requirements of different norm approximationschemes are not considered for this particular comparison ofcomputational complexity. It is also important to note that in theproposed sortsum approximation, five adders may be required for betterapproximation accuracy, while four adders may be sufficient to achieve aslightly worse performance.

Exemplary Simulation Results

Simulation results are presented in this disclosure for differentapproximate algorithms for computation of l₂ norms. FIG. 8 showsrecommended values of tuning factors and corresponding mean squareerrors (MSE) at a signal-to-noise ratio (SNR) of 0 dB for different normapproximation schemes. As an illustrative example, a wireless systemwith two transmit/receive antennas may be considered, Additive WhiteGaussian Noise (AWGN) channels may be assumed, and 64-QAM modulation maybe applied at the transmitter.

FIGS. 9-11 show error rate performance loss in dB units of theMax-Log-MAP PC-MLD with different approximate techniques for computationof l₂ norms in Pedestrian A channels where the speed of mobilesubscriber is 3 km/h, while the 64-QAM, 16-QAM and QPSK modulations arerespectively applied, and the packet error rate of 10-2 is assumed.50,000 data packets are used for simulations, tailbiting convolutionalcodes (TBCC) of rate 1/2, 2/3 and 3/4 and/or convolutional Turbo codes(CTC) of rate 1/2, 2/3 and 5/6 can be applied. It is assumed perfectchannel knowledge at the receiver.

The following schemes are evaluated by simulations: the Max-Log-MAPPC-ML detection with exact l₂ norm calculation, the Max-Log-MAP PC-MLdetection with the proposed sortsum approximate algorithm with tuningfactor F=[0875 04375 0.3125 0.125], and the Max-Log-MAP PC-ML detectionwith the proposed maxsum approximate algorithm with tuning factor of0.25. It can be observed from FIGS. 9-11 that the averaged performanceloss considering all code rates and modulation types is only about0.01˜0.03 dB for the sortsum approximation and about 0.05˜0.09 dB forthe maxsum approximation compared to the exact l₂ norm calculationapproach.

The various operations of methods described above may be performed byvarious hardware and/or software component(s) and/or module(s)corresponding to means-plus-function blocks illustrated in the Figures.For example, blocks 610-670 illustrated in FIG. 6 correspond tomeans-plus-function blocks 610A-670A illustrated in FIG. 6A. Moregenerally, where there are methods illustrated in Figures havingcorresponding counterpart means-plus-function Figures, the operationblocks correspond to means-plus-function blocks with similar numbering.

The various illustrative logical blocks, modules and circuits describedin connection with the present disclosure may be implemented orperformed with a general purpose processor, a digital signal processor(DSP), an application specific integrated circuit (ASIC), a fieldprogrammable gate array signal (FPGA) or other programmable logic device(PLD), discrete gate or transistor logic, discrete hardware componentsor any combination thereof designed to perform the functions describedherein. A general purpose processor may be a microprocessor, but in thealternative, the processor may be any commercially available processor,controller, microcontroller or state machine. A processor may also beimplemented as a combination of computing devices, e.g., a combinationof a DSP and a microprocessor, a plurality of microprocessors, one ormore microprocessors in conjunction with a DSP core, or any other suchconfiguration.

The steps of a method or algorithm described in connection with thepresent disclosure may be embodied directly in hardware, in a softwaremodule executed by a processor, or in a combination of the two. Asoftware module may reside in any form of storage medium that is knownin the art. Some examples of storage media that may be used includerandom access memory (RAM), read only memory (ROM), flash memory, EPROMmemory, EEPROM memory, registers, a hard disk, a removable disk, aCD-ROM and so forth. A software module may comprise a singleinstruction, or many instructions, and may be distributed over severaldifferent code segments, among different programs, and across multiplestorage media. A storage medium may be coupled to a processor such thatthe processor can read information from, and write information to, thestorage medium. In the alternative, the storage medium may be integralto the processor.

The methods disclosed herein comprise one or more steps or actions forachieving the described method. The method steps and/or actions may beinterchanged with one another without departing from the scope of theclaims. In other words, unless a specific order of steps or actions isspecified, the order and/or use of specific steps and/or actions may bemodified without departing from the scope of the claims.

The functions described may be implemented in hardware, software,firmware or any combination thereof. If implemented in software, thefunctions may be stored as one or more instructions on acomputer-readable medium. A storage media may be any available mediathat can be accessed by a computer. By way of example, and notlimitation, such computer-readable media can comprise RAM, ROM, EEPROM,CD-ROM or other optical disk storage, magnetic disk storage or othermagnetic storage devices, or any other medium that can be used to carryor store desired program code in the form of instructions or datastructures and that can be accessed by a computer. Disk and disc, asused herein, include compact disc (CD), laser disc, optical disc,digital versatile disc (DVD), floppy disk, and Blu-Ray® disc where disksusually reproduce data magnetically, while discs reproduce dataoptically with lasers.

Software or instructions may also be transmitted over a transmissionmedium. For example, if the software is transmitted from a website,server, or other remote source using a coaxial cable, fiber optic cable,twisted pair, digital subscriber line (DSL), or wireless technologiessuch as infrared, radio, and microwave, then the coaxial cable, fiberoptic cable, twisted pair, DSL, or wireless technologies such asinfrared, radio, and microwave are included in the definition oftransmission medium.

Further, it should be appreciated that modules and/or other appropriatemeans for performing the methods and techniques described herein can bedownloaded and/or otherwise obtained by a user terminal and/or basestation as applicable. For example, such a device can be coupled to aserver to facilitate the transfer of means for performing the methodsdescribed herein. Alternatively, various methods described herein can beprovided via storage means (e.g., RAM, ROM, a physical storage mediumsuch as a compact disc (CD) or floppy disk, etc.), such that a userterminal and/or base station can obtain the various methods uponcoupling or providing the storage means to the device. Moreover, anyother suitable technique for providing the methods and techniquesdescribed herein to a device can be utilized.

It is to be understood that the claims are not limited to the preciseconfiguration and components illustrated above. Various modifications,changes and variations may be made in the arrangement, operation anddetails of the methods and apparatus described above without departingfrom the scope of the claims.

1. A method for decoding data transmitted on a plurality of spatialchannels in a multiple-input multiple-output wireless communicationssystem, comprising: calculating non-squared l₂ norm metrics for a set ofhypothesized symbols that correspond to coded bits transmitted over thespatial channels; searching the non-squared l₂ norm metrics for optimummetrics; squaring optimum metrics found during the searching to obtainpost-squared metrics; and calculating log-likelihood ratio values forcoded bits transmitted over the spatial channels using post-squaredmetrics.
 2. The method of claim 1, wherein calculating un-squared l₂norm metrics for a set of hypothesized symbols that correspond to codedbits transmitted over the spatial channels comprises: utilizingapproximated l₂ norm values.
 3. The method of claim 1 furthercomprising: computing approximate l₂ norm values using an approximationtechnique selected from a group of approximation techniques, the groupof approximation techniques including tri-maxmin approximation, sortsumapproximation, and maxsum approximation.
 4. The method of claim 3further comprising: tuning at least one factor that affects an accuracyof the approximation technique.
 5. The method of claim 1, whereinsearching the un-squared l₂ norm metrics for optimum metrics comprises:searching the non-squared l₂ norm metrics for minima.
 6. The method ofclaim 1 further comprising: permuting a matrix of channel estimates forthe spatial channels, to generate permuted matrices of channel estimatesfor the plurality of spatial streams; and performing QR decomposition ofthe permuted matrices to generate a unitary matrix and an uppertriangular matrix for the plurality of spatial streams.
 7. The method ofclaim 6 further comprising: rotating a received signal with the unitarymatrices to generate filtered outputs; slicing filtered outputs toobtain sliced coded bits for the plurality of spatial streams;hypothesizing symbols that may be transmitted on the spatial streamsusing sliced coded bits; and decoding bits transmitted over theplurality of spatial sub-channels by using calculated log-likelihoodratios of coded bits.
 8. An apparatus for decoding data transmitted on aplurality of spatial channels in a multiple-input multiple-outputwireless communications system, comprising: logic for calculatingnon-squared l₂ norm metrics for a set of hypothesized symbols thatcorrespond to coded bits transmitted over the spatial channels; logicfor searching the non-squared l₂ norm metrics for optimum metrics; logicfor squaring optimum metrics found during the searching to obtainpost-squared metrics; and logic for calculating log-likelihood ratiovalues for coded bits transmitted over the spatial channels usingpost-squared metrics.
 9. The apparatus of claim 8, wherein the logic forcalculating un-squared l₂ norm metrics for a set of hypothesized symbolsthat correspond to coded bits transmitted over the spatial channelscomprises: logic for utilizing approximated l₂ norm values.
 10. Theapparatus of claim 8 further comprising: logic for computing approximatel₂ norm values using an approximation technique selected from a group ofapproximation techniques, the group of approximation techniquesincluding tri-maxmin approximation, sortsum approximation, and maxsumapproximation.
 11. The apparatus of claim 10 further comprising: logicfor tuning at least one factor that affects an accuracy of theapproximation technique.
 12. The apparatus of claim 8, wherein the logicfor searching the un-squared 1 ₂ norm metrics for optimum metricscomprises: logic for searching the non-squared l₂ norm metrics forminima.
 13. The apparatus of claim 8 further comprising: logic forpermuting a matrix of channel estimates for the spatial channels, togenerate permuted matrices of channel estimates for the plurality ofspatial streams; and logic for performing QR decomposition of thepermuted matrices to generate a unitary matrix and an upper triangularmatrix for the plurality of spatial streams.
 14. The apparatus of claim13 further comprising: logic for rotating a received signal with theunitary matrices to generate filtered outputs; logic for slicingfiltered outputs to obtain sliced coded bits for the plurality ofspatial streams; logic for hypothesizing symbols that may be transmittedon the spatial streams using sliced coded bits; and logic for decodingbits transmitted over the plurality of spatial sub-channels by usingcalculated log-likelihood ratios of coded bits.
 15. An apparatus fordecoding data transmitted on a plurality of spatial channels in amultiple-input multiple-output wireless communications system,comprising: means for calculating non-squared l₂ norm metrics for a setof hypothesized symbols that correspond to coded bits transmitted overthe spatial channels; means for searching the non-squared l₂ normmetrics for optimum metrics; means for squaring optimum metrics foundduring the searching to obtain post-squared metrics; and means forcalculating log-likelihood ratio values for coded bits transmitted overthe spatial channels using post-squared metrics.
 16. The apparatus ofclaim 15, wherein the means for calculating un-squared l₂ norm metricsfor a set of hypothesized symbols that correspond to coded bitstransmitted over the spatial channels comprises: means for utilizingapproximated l₂ norm values.
 17. The apparatus of claim 15 furthercomprising: means for computing approximate l₂ norm values using anapproximation technique selected from a group of approximationtechniques, the group of approximation techniques including tri-maxminapproximation, sortsum approximation, and maxsum approximation.
 18. Theapparatus of claim 17 further comprising: means for tuning at least onefactor that affects an accuracy of the approximation technique.
 19. Theapparatus of claim 15, wherein the means for searching the un-squared l₂norm metrics for optimum metrics comprises: means for searching thenon-squared l₂ norm metrics for minima.
 20. The apparatus of claim 15further comprising: means for permuting a matrix of channel estimatesfor the spatial channels, to generate permuted matrices of channelestimates for the plurality of spatial streams; and means for performingQR decomposition of the permuted matrices to generate a unitary matrixand an upper triangular matrix for the plurality of spatial streams. 21.The apparatus of claim 20 further comprising: means for rotating areceived signal with the unitary matrices to generate filtered outputs;means for slicing filtered outputs to obtain sliced coded bits for theplurality of spatial streams; means for hypothesizing symbols that maybe transmitted on the spatial streams using sliced coded bits; and meansfor decoding bits transmitted over the plurality of spatial sub-channelsby using calculated log-likelihood ratios of coded bits.
 22. Acomputer-program product for decoding data transmitted on a plurality ofspatial channels in a multiple-input multiple-output wirelesscommunications system, comprising a non-transitory computer readablemedium having instructions stored thereon, the instructions beingexecutable by one or more processors and the instructions comprising:instructions for calculating non-squared l₂ norm metrics for a set ofhypothesized symbols that correspond to coded bits transmitted over thespatial channels; instructions for searching the non-squared l₂ normmetrics for optimum metrics; instructions for squaring optimum metricsfound during the searching to obtain post-squared metrics; andinstructions for calculating log-likelihood ratio values for coded bitstransmitted over the spatial channels using post-squared metrics. 23.The computer-program product of claim 22, wherein the instructions forcalculating un-squared l₂ norm metrics for a set of hypothesized symbolsthat correspond to coded bits transmitted over the spatial channelsfurther comprise: instructions for utilizing approximated l₂ normvalues.
 24. The computer-program product of claim 22, wherein theinstructions further comprise: instructions for computing approximate l₂norm values using an approximation technique selected from a group ofapproximation techniques, the group of approximation techniquesincluding tri-maxmin approximation, sortsum approximation, and maxsumapproximation.
 25. The computer-program product of claim 24, wherein theinstructions further comprise: instructions for tuning at least onefactor that affects an accuracy of the approximation technique.
 26. Thecomputer-program product of claim 22, wherein the instructions forsearching the un-squared l₂ norm metrics for optimum metrics furthercomprise: instructions for searching the non-squared l₂ norm metrics forminima.
 27. The computer-program product of claim 22, wherein theinstructions further comprise: instructions for permuting a matrix ofchannel estimates for the spatial channels, to generate permutedmatrices of channel estimates for the plurality of spatial streams; andinstructions for performing QR decomposition of the permuted matrices togenerate a unitary matrix and an upper triangular matrix for theplurality of spatial streams.
 28. The computer-program product of claim27, wherein the instructions further comprise: instructions for rotatinga received signal with the unitary matrices to generate filteredoutputs; instructions for slicing filtered outputs to obtain slicedcoded bits for the plurality of spatial streams; instructions forhypothesizing symbols that may be transmitted on the spatial streamsusing sliced coded bits; and instructions for decoding bits transmittedover the plurality of spatial sub-channels by using calculatedlog-likelihood ratios of coded bits.